Canonical Basis

linear-algebra

Definition

Canonical Basis

Let $V$ be a $n$-dimensional vector space. The canonical basis $\mathcal{B}$ defined as:

$$\mathcal{B}=\{e_1,e_2,\dots ,e_n\}$$

where each vector $e_j$ has a $1$ in the $j$-th position and $0$ elsewhere:

$$e_1=\left[\begin{array}{c}1\\ 0\\ ⋮\\ 0\end{array}\right],e_2=\left[\begin{array}{c}0\\ 1\\ ⋮\\ 0\end{array}\right],\dots ,e_n=\left[\begin{array}{c}0\\ 0\\ ⋮\\ 1\end{array}\right]$$

Kronecker Delta

Let $(e_j{)}_i$ denote the $i$-th scalar component of the vector $e_j$. Then:

$$(e_j{)}_i={\delta }_{ij}$$

where ${\delta }_{ij}$ is the Kronecker Delta.

Orthonormal Basis

The canonical basis is also an orthonormal basis under the usual dot product as it is tightly related to the Kronecker Delta. Observe that the column vectors of the $n$-identity matrix is equal to the set of the standard basis:

$$I_n=\left[\begin{array}{ccccc}1& 0& \dots & 0& 0\\ 0& 1& \dots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \dots & 1& 0\\ 0& 0& \dots & 0& 1\end{array}\right]=\left[\begin{array}{cccc}{e}_1& e_2& \dots & e_n\end{array}\right]$$